Why do we write yours sincerely instead of your sincerely. Here I think before sincerely we need a possessive adjective and 'your' is doing this job well.
"Yours sincerely" is short for "I am yours sincerely". "Sincerely" is always an adverb. Because there is no noun, "yours" is used instead of "your".
(If signing off of letters like this seems weird, it was definitely a lot weirder in the past.)